1. Three-Phase Systems Professor Mohamed A. El-Sharkawi
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2. El-Sharkawi@University of Washington

3. El-Sharkawi@University of Washington

4. El-Sharkawi@University of Washington

5. El-Sharkawi@University of Washington
Single Phase
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AC Waveform
One
Cycle
Voltage or Current
Peak
Maximum
Time
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ac Waveform
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8. El-Sharkawi@University of Washington
AC Representation w t v
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9. El-Sharkawi@University of Washington
How AC is Generated
Stator N S
Rotor
Windings
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10. El-Sharkawi@University of WashingtonX N S  ϕ
900
2700
Angle
1800
3600
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11. El-Sharkawi@University of Washington
Alternating Current v N S N S N S N S . N S N S N S N S N S
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12. El-Sharkawi@University of Washington
Reference V2 w t v 1 2 q
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13. El-Sharkawi@University of Washington
Reference V2 w t v 1 2 q
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14. v i Voltage and Current can be Out of Phase
load V I + _ i v q w t V I
is the phase shift of current
also known as the power factor angle
It is due to the presence of inductive and capacitive elements.
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15. El-Sharkawi@University of Washington
Example
If V=120 V and Z=4 + j3 , calculate the current and power factor. V Z I + _
Lagging V I
Notice that the pf angle is the angle of the impedance
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16. El-Sharkawi@University of Washington
Polarities Z I + _
Load V I + _
Source
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17. El-Sharkawi@University of Washington
Three-Phase
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18. El-Sharkawi@University of Washington
Three-Phase va N S N S N S N S N S N S N S N S I b vb I a vc I c
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19. Three-Phase Generator
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20. El-Sharkawi@University of Washington
Three-phase system v
aa'
bb'
cc'
Reference
120o
vaa’
vbb’
vcc’
120o
120o
Time
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21. Why do we use 3-phase systems?
Three-phase system produces rotating magnetic field.
Three-phase motors can start without the need for extra equipment.
For the same physical size, a three-phase generator produces more power than a single phase generator.
Three-phase lines transmit more power.
Three-phase lines are more reliable.
In distribution circuit, you can operate the system with one missing phase.
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22. c b a ¢ ¢ c b ¢ a How is Three-Phase System Connected? X X X
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23. How is Three-Phase System Connected?
Any three-phase generator has 6 terminal wires
Transmitting 6 wire over a long distance is expensive
Instead, three-wire system is used by connecting the six wires as Y or Delta
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24. El-Sharkawi@University of Washington
Y-Connected Source a n c b X X c b X a
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Y-Connected Source
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Y-Connected Source c c v cn v an
Reference n a v bn
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27. Delta-Connected Sourceb X X c b X a
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28. Delta () Connection: Source
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32. Phase and LineVoltages
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33. El-Sharkawi@University of Washingtonc
Phase voltage v cn v an
Reference a n a b v bn c b v an v bn v cn n
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34. El-Sharkawi@University of Washingtoncn c
Line-to-line voltage v bc v ca n v an a n c b a
Reference v ca v ab v bn b v ab v bc
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35. El-Sharkawi@University of Washingtonca n v bn cn an v ab -v bn
Reference v bc
Reference v ab ca bc bn b n an a cn c
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Keep in mind
Unless stated in the problems, the following assumptions are to be used:
All voltages are line-to-line quantities
All powers are for the three phases
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37. El-Sharkawi@University of Washington
Example
Let v ab
Calculate the line-to-line voltage Vab v an
Reference
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Main Conclusions
Line-to-line voltage is greater than phase voltage by
Line-to-line voltage leads phase voltage by
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39. El-Sharkawi@University of Washington
Y-Load connection
Load
Most loads are connected in Y
To have access to the ground potential
To balance the voltage across the load
Each load impedance (Z) is called load impedance
Voltage across the load impedance is called load voltage
Load current a +
Three-phase feeder I Z a I b n Z Z + + b I c
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40. Main Conclusions for Y-Connected Load
Load voltage is equal to the Phase voltage
Line current is equal to load current by
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41. Delta-Load connection
Mostly in industrial loads
No Access to ground
Used to increase voltage across the load
Each load impedance (Z) is called load impedance
Voltage across the load impedance is called load voltage
Load current a - +
Three-phase feeder
Ica Z
Iab Z - + c Z b - +
Ibc
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42. Y-Connection Source and Load
Line current
Transmission
Line
Source
Load I
Load current a a a + + I Z V I a an a I n b n V V Z Z cn bn + I c + + b c + b I I I b c c b I c
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43. El-Sharkawi@University of Washington
For balanced system
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44. El-Sharkawi@University of Washingtoncn I c q n V an
Reference I b q V bn I a q
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45. Neutral Current of Balanced LoadI c
For balanced system I a I b
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46. Single-Phase RepresentationIa Va Va Z Ia + _
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47. Delta () Connection Source and Load
Line current
Transmission
Line
Source
Load
Phase current I
Load current a a a - + - +
Ica
Ica Z
Iab
Iab Z V ca V - ab + - + I c Z b c b b - + V bc
Ibc - +
Ibc I c
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48. Delta () Connection: SourceIa a _
Ica +
Ia is line current
Iab is generator current
Vca
Vab
Iab _ Ib + b
Vbc c _ +
Ibc Ic
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49. El-Sharkawi@University of Washington
Kirchhoff’s Current law at node a
Transmission
Line
Load I
Load current c c - +
Ibc
Ica
Reference v bc ab ca a b c Z Z - q + I b Z a a - +
Iab I q b q
Voltage Diagram
Vab is chosen reference
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50. El-Sharkawi@University of Washington
Line Currents
Reference v bc ab ca a b c V ab
Refere q I ab q I ca I bc q q
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Line Currents V ab
Refere q I ab
300 I ca
-Ica I a
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52. El-Sharkawi@University of Washington
Example
Iab V ab q
300
Let
Reference
100 I
Calculate the phase current Iab of a delta
circuit a
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53. Main Conclusions for Delta-Connected Load
Load voltage is equal to the line-to-line voltage
Line current is greater than load current by
Line current lags the load current by
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54. El-Sharkawi@University of Washington
Mixed Connection
Line current
Transmission
Line
Source
Load
Phase current I
Load current a a a + I Z
Ica a
Iab V ca V I ab b n Z Z - I c b c b I b c V bc
Ibc I c
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55. El-Sharkawi@University of Washington
Y- Transformation a b c Ia Ib Ic Z a b c Ia Ib Ic ZY
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Y- Transformation
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57. El-Sharkawi@University of Washington
Y- Transformation a b c Ia Ib Ic Z a b c Ia Ib Ic ZY
THEN
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58. El-Sharkawi@University of Washington
Example
1. Calculate the current of the load
2. Calculate the equivalent Y load
3. Calculate the load current of the equivalent Y load b c Ia Ic
Van = 120 v a Ib
Z = 4 + j 3
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59. El-Sharkawi@University of Washingtonb c Ia Ic
Van Ib Z
Part 1:
Part 2:
Part 3: The load current in  load is equal to
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60. El-Sharkawi@University of Washington
Example
Calculate the line current
Z2 = j 9
Z1 = 4 + j 3 Ia a b c
Van = 120 v c b b c Ic
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61. El-Sharkawi@University of Washington
Change delta to Y
Z2y = 4 - j 3
Z1 = 4 + j 3 Ia a b c
Van = 120 v b b c c Ic
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62. El-Sharkawi@University of Washington
Zeq = 25/8  Ia a b c
Van = 120 v b c Ic
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63. Instantaneous Electric Power [p(t)]V I
Fixed average
Zero average
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64. Instantaneous Electric Power [p(t)]
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Real Power (Average P)
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66. El-Sharkawi@University of Washington
Real Power (Average P) p wt
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Reactive Power [Q]
Important points!
Frequency of h(t) is double the
frequency of supply
voltage
Average value of h(t) is
zero w t
h(t) q p
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Real power
Reactive power
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69. El-Sharkawi@University of Washington
Complex Power (S)
IMPORTANT
 is the power factor angle V I
Real Power
Reactive Power
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70. Power of 3-phase circuits
Iphase
Vphase
For Single phase
IMPORTANT
 is the angle between
phase voltage and
phase current.
Use voltage as a
reference
For 3-phase
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71. Real Power in Delta Circuitb c Ia Ib Ic
Iab
Ica
Ibc + _
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72. Reactive Power in Delta Circuitb c Ia Ib Ic
Iab
Ica
Ibc + _
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73. El-Sharkawi@University of Washington
Real Power in Y Circuit a b c Ia Ib Ic Va Vc Vb
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74. Reactive Power in Y Circuitb c Ia Ib Ic Va Vc Vb
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75. El-Sharkawi@University of Washington
Example
Calculate the load power b c Ia Ic
Van = 120 v a Ib
Z = 4 - j 3
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76. El-Sharkawi@University of Washingtonb c Ia Ic
Van Ib Z V ab I ca V an
300 q n I ab I bc
300 I a
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77. El-Sharkawi@University of Washington
Method #1
Method #2
Keep in mind
The pf angle is the angle of the load impedance.
It is the same as the angle between the phase voltage and the
phase current
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